CAVALIER!, RUGNAVENTURA, was born at Milan in 1593, and entered into the order of Jesuits at an early age. lie was afterwards professor at Bologna, and died there December 3, 1617. He was a pupil of Galileo, according to the of his friend Riccioli, who prefeeess himself much indebted to his &matinee in his studies. But it is said that he did not obtain the professorship at Bologna, in spite of the strong recommeodetion of Galileo, until his skill in astrology bad been duly on-titled. lie was a victim to the gout both in hand sad foot, and was confined almost entirely to his bed for twelvo years before his death. There is an eloge of him by Friel (quoted by Lacroix, ' Biog. Univ..). All his works were published at Bologna, 2-eing 'Speochio Ustorio; &c. 1632; ' Directorium Generale Urana enetricum,' 1632; 'Geometria Indivisibilibus Continuer= &e. 1635; 'Trigonometria Plana at Spherics,' 1635; 'Rota Planetaria,' 1640; Geometries Sex,' 1647.
If we may judge from a contemporary biographer, Ghilini, ' Teatro d'Uosaird Lettersti,' Cavilled (or Cavalerius, as his name is usually Wished) must have enjoyed • remarkable reputation in his day. But he has descended to posterity solely through his method of ambrisitia, one of the predecessors of the doctrine of fluxion', and which Plasnow, LTIRXITZ ; DITTERENTIAL CALCULUS in ARTS AHD SOLXCIISI DIY.) must be considered as one of the first attempts at au orris:need method of dealing with the difficulties of the solution of which Archimedes had given the first example. Cavalieri considers a limo as composed of an infinite number of points, • surface of an infinite number of lines, and so on, as in the following sentence: "dune manifestum set figural planes noble ad hater telex parallelia fills contexts coecipiendas ease; solid* vero ad instar librorum, qui paralidis foilE eoacervantur. Cum vero in tele aunt semper file, et In libris suoper foils numero finite, habet [habent) enim aliquam rravritietn, noble in figuria plaids line.., In solidi' vero plane numero indefinite . . . supponenda runt: This method, absolutely considered, is defective and even erroneous, but the error is of the isms kind as that of Leibnitz, who considered • curve as composed of an infinite number of infinitely wall chords, and • endue of Infinitely small rectangles. The error in both I. one which does not affect the result, for this
reason, that it consists In using the simplifying effect of a certain supposition too early in the process, by which the logic of the inves tigatioa may be injured, but the result is not affected. For instance, Cavalerins would consider a right-angled triangle as follows. Let n he the number of points in the base, then the perpendiculars at these points are in arithmetical proareseion, 0, a, 2 a, &a. . . . no : the suns of all of which is (le 1) a, or 8 a. ma, throwing away 4 na as locotudderable compared with 4N. na, when n Is infinite. But 8 ma ea Is simply 4 taws it perpendicular. Compare this method (absurd and almost unintelligible as it is, In the literal sense of the terms) with the following. Divide the base (6) into et equal parts, each of which is therefore ;. Let the rcrpendicular be p, consequently the p perpendicular. at the extremities of the parts are -0 — np to — n ' and each multiplied by —, and the cam of the whole being taken, we have— /p IspN bp by / I J="i = 2 0 + ;i.
This le the sum of the inscribed rectangles, which approaches without limit to the area of the triangle as n Increased without limit. But it approaches to i 6 p on the same supposition • whence 4 6 p is the area of the triangle. Either method, with might be made to give true moult., and in an Intelligible manner; but that of Cavalerius is very subject to error, and, we may say, requires a knowledge of bettor methods to understand It. But it is nevertheless the first attempt at geeeralisation, and serves to illustrate the position main tained by us (Itaaaow), that neither the Newton nor the infinitesimal. of Leibuits were the actual methods by means of which the Differential Calculus (as now known) was madepowerfuL Cavalerium, will, the ' methods of development ' of Newton, might have eatablithed his title to the invention. But his algebra was very imperfect, even for his day : we cannot see proof, in 1647, that he had ever seen the writing. of Vista, who died in 1603. The celebrated Guldinus wrote against the method of indivisible, and was answered by Cavalerius in the third of the Exercitationes Geometricx.' Roberval claimed the method as his own, but his first publications on the subject followed those of Cavalerius.